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## Application of Derivatives Class 12 MCQs Questions with Answers

Don’t forget to practice the multitude of MCQ questions on Application of Derivatives Class 12 with answers so you can apply your skills during the exam.

Question 1.

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is:

(a) 10π

(b) 12π

(c) 8π

(d) 11π

## Answer

Answer: (b) 12π

Question 2.

The total revenue received from the sale of x units of a product is given by R (x) = 3x² + 36x + 5. The marginal revenue, when x = 15 is:

(a) 116

(b) 96

(c) 90

(d) 126.

## Answer

Answer: (d) 126.

Question 3.

The interval in which y = x² e^{-x} is increasing with respect to x is:

(a) (-∞, ∞)

(b) (-2,0)

(c) (2, ∞)

(d) (0, 2).

## Answer

Answer: (d) (0, 2).

Question 4.

The slope of the normal to the curve y = 2x² + 3 sin x at x = 0 is

(a) 3

(b) \(\frac { 1 }{3}\)

(c) -3

(d) –\(\frac { 1 }{3}\)

## Answer

Answer: (d) –\(\frac { 1 }{3}\)

Question 5.

The line y = x + 1 is a tangent to the curve y² = 4x at the point:

(a) (1, 2)

(b) (2, 1)

(c) (1, -2)

(d) (-1, 2).

## Answer

Answer: (a) (1, 2)

Question 6.

If f(x) = 3x² + 15x + 5, then the approximate value of f(3.02) is:

(a) 47.66

(b) 57.66

(c) 67.66

(d) 77.66.

## Answer

Answer: (d) 77.66.

Question 7.

The approximate change in the volume of a cube of side x metres caused by increasing the side by 3% is:

(a) 0.06 x³ m³

(b) 0.6 x³ m³

(c) 0.09 x³m³

(d) 0.9 x³ m³

## Answer

Answer: (c) 0.09 x³m³

Question 8.

The point on the curve x² = 2y, which is nearest to the point (0, 5), is:

(a) (2 √2, 4)

(b) (2 √2, 0)

(c) (0, 0)

(d) (2, 2).

## Answer

Answer: (a) (2 √2, 4)

Question 9.

For all real values of x, the minimum value of \(\frac { 1-x+x^2 }{1+x+x^2}\) is

(a) 0

(b) 1

(c) 3

(d) \(\frac { 1 }{3}\)

## Answer

Answer: (d) \(\frac { 1 }{3}\)

Question 10.

The maximum value of [x (x – 1) + 1]^{1/3}, 0 ≤ x ≤ 1 is

(a) (\(\frac { 1 }{3}\))^{\(\frac { 1 }{3}\)}

(b) \(\frac { 1 }{2}\)

(c) 1

(d) 0

## Answer

Answer: (c) 1

Question 11.

A cylindrical tank of radius 10 mis being filled with wheat at the rate of 314 cubic m per minute. Then the depth of the wheat is increasing at the rate of:

(a) 1 m/minute

(b) 0 × 1 m/minute

(c) 1 × 1 m/minute

(d) 0 × 5 m/minute.

## Answer

Answer: (a) 1 m/minute

Question 12.

The slope of the tangent to the curve x = t² + 3t – 8, y = 2 t² – 2t – 5 at the point (2, -1) is:

(a) \(\frac { 22 }{7}\)

(b) \(\frac { 6 }{7}\)

(c) \(\frac { 7 }{6}\)

(d) \(\frac { -6 }{7}\)

## Answer

Answer: (b) \(\frac { 6 }{7}\)

Question 13.

The line y = mx + 1 is a tangent to the curve y² = 4x if the value of m is:

(a) 1

(b) 2

(c) 3

(d) \(\frac { 1 }{2}\)

## Answer

Answer: (a) 1

Question 14.

The normal at the point (1, 1) on the curve 2y + x² = 3 is

(a) x + y = 0

(b) x – y = 0

(c) x + y + 1 = 0

(d) x – y + 1 = 0.

## Answer

Answer: (b) x – y = 0

Question 15.

The normal to the curve x² = 4y passing through (2, 1) is:

(a) x + y = 3

(b) x – y = 3

(c) x + y = 1

(d) x – y = 1.

## Answer

Answer: (a) x + y = 3

Question 16.

The points on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes are

(a) (4, ±\(\frac { 8 }{3}\))

(b) (4, –\(\frac { 8 }{3}\))

(c) (4, ±\(\frac { 3 }{8}\))

(d) (±4, \(\frac { 3 }{8}\))

## Answer

Answer: (a) (4, ±\(\frac { 8 }{3}\))

Question 17.

The abscissa of the point on the curve 3y = 6x – 5x³, the normal at which passes through origin is:

(a) 1

(b) \(\frac { 1 }{3}\)

(c) 2

(d) \(\frac { 1 }{2}\)

## Answer

Answer: (a) 1

Question 18.

The two curves x³ – 3xy² + 2 = 0 and 3x²y – y³ = 2

(a) touch each other

(b) cut at right angle

(c) cut at an angle \(\frac { π }{3}\)

(d) cut at an angle \(\frac { π }{4}\)

## Answer

Answer: (b) cut at right angle

Question 19.

The tangent to the curve given by:

x = e^{t} cos t, y = e^{t} sm t at t = \(\frac { π }{4}\) makes with x-axis an angle:

(a) 0

(b) \(\frac { π }{4}\)

(c) \(\frac { π }{3}\)

(d) \(\frac { π }{2}\)

## Answer

Answer: (d) \(\frac { π }{2}\)

Question 20.

The equation of the normal to the curve y = sin x at (0, 0) is

(a) x = 0

(b) y = 0

(c) x + y = 0

(d) x – y = 0.

## Answer

Answer: (c) x + y = 0

Question 21.

The point on the curve y² = x, where the tangent makes an angle of \(\frac { π }{4}\) with x-axis is:

(a) (\(\frac { 1 }{2}\), \(\frac { 1 }{4}\))

(b) (\(\frac { 1 }{4}\), \(\frac { 1 }{2}\))

(c) (4, 2)

(d) (1, 1).

## Answer

Answer: (b) (\(\frac { 1 }{4}\), \(\frac { 1 }{2}\))

Question 22.

Let f: R → R be a positive increasing function with:

\( \lim _{x \rightarrow \infty}\) \(\frac { f(3x) }{f(x)}\) = 1. Then \( \lim _{x \rightarrow \infty}\) \(\frac { f(2x) }{f(x)}\) =

(a) 1

(b) \(\frac { 2 }{3}\)

(c) \(\frac { 3 }{2}\)

(d) 3.

## Answer

Answer: (a) 1

Hint:

Since f(x) is a positive increasing function

∴ 0 < f(x) < f(2x) < f(3x)

Question 23.

The real number k for which the equation 2x³ + 3x + k = 0 has two distinct real roots in [0,1]:

(a) lies between 2 and 3

(b) lies between -1 and 0

(c) does not exist

(d) lies between 1 and 2.

## Answer

Answer: (c) does not exist

Hint:

If 2x³ + 3x + k = 0 has two distinct real roots in [0, 1], then f'(x) will change sign.

But f'(x) = cx² + 3 > 0

Hence, no value of k exists.

Question 24.

If f and g are differentiable functions on [0, 1] satisfying f(0) = 2 = g(l), g(0) = 0 and f(1) = 6, then for some c ∈ ] 0, 1 [:

(a) 2f'(c) = 3g'(c)

(b) f'(c) = g'(c)

(c) f'(c) = 2g'(c)

(d) 2f'(c) = g'(c).

## Answer

Answer: (c) f'(c) = 2g'(c)

Hint:

Let h(x) = f(x) – 2g(x).

∴ h'(x) =f'(x) – 2g'(x).

Here h(0) = h(1) = 2.

By Rolle’s Theorem, h’ (c) = 0

⇒ f'(c) = 2g'(c).

Question 25.

Twenty metres of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower bed is:

(a) 25

(b) 30

(c) 12.5

(d) 10.

## Answer

Answer: (a) 25

Hint:

Total length = r + r + rθ = 20

∴ r = 5 gives max-area.

Hence, from (2) maximum area,

A =10(5) – 25 = 25.

Question 26.

Let f (x) = x² – \(\frac { 1 }{x^2}\) and g(x) = x – \(\frac { 1 }{x}\), x ∈ R – {-1, 0, 1}. If h(x) = \(\frac { f(x) }{g(x)}\), then the local minimum value of h(x) is:

(a) 3

(b) -3

(c) -2√2

(d) 2√2.

## Answer

Answer: (d) 2√2.

Hint:

Fill in the blanks

Question 1.

Rate of change of the area of a circle with respect to its radius when r = 4 cm is ……………..

## Answer

Answer: 8π cm²/m.

Question 2.

Rate of change of the volume of a ball with respect to its radius is ………………

## Answer

Answer: 4πr²

Question 3.

The function f(x) = |x| is strictly ……………… in (0, ∞)

## Answer

Answer: Increasing.

Question 4.

Logarithmic function is strictly ………………. in (0, ∞).

## Answer

Answer: Increasing.

Question 5.

The value of ‘a’ for which f(x) = sin x – ax + b is decreasing function on R is ……………..

## Answer

Answer: ≤ 1.

Question 6.

Slope of the tangent to the curve x = at², y = 2t at t = 2 is ……………..

## Answer

Answer: \(\frac { 1 }{2}\)

Question 7.

If tangent to the curve y² + 3x – 7 = 0 at the point (h, k) is parallel to like x – y – 4, then the value of k is ……………….

## Answer

Answer: –\(\frac { 3 }{2}\)

Question 8.

For the curve y = 5x – 2x³, if x increases at the tangents of 2 units/sec; then at x = 3 the slope of the curve is changes at ……………..

## Answer

Answer: decreasing at the rate of 72 units/sec.

Question 9.

If x > 0, y > 0 and xy = 5, then the minimum value of x + y is ………………

## Answer

Answer: 10.

Question 10.

Maximum value of:

f(x) = – (x – 1)² + 2 is ………………

## Answer

Answer: 2.

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